WWIFN is excellent as ever.
I think a few tough words are needed here with H.
Quite simply, it is HIS job to square his movements with her, not yours solost You need to hold firm, he said to you to let him know if ever you needed childcare, you know he was keen to do it, you asked he said yes.
TBH, if he is free to do it, they are his DC and he can make that decision. BB doesn't factor into it. Stupid Cow, if she relaxed and STFU about it, she'd BE more involved, but the way she is being now, she is forcing everyone to isolate her and contain her away from the family relationship dynamic.
IMHO you need to tell H in no uncertain terms that you have absolutely no interest whatsoever what her opinion is on anything to do with you DC, she simply has no role there.
Tell him that he is not to ring you and wriggle out of commitments he has made with regard to the care of his DC, because of her insecurity. Simply, you do not want to hear FROM her, or ABOUT her. Tell him in no uncertain terms that you do not welcome texts from women such as her and that she needs to stick with her own kind.
If you want to arrange access with H to see his DC and vice versa, literally no-one will filter that, no-one will get in the way of it and H is big enough and bright enough to manage his OWN calendar.
The innocent fool image is bang on, I see him in short trousers simpering and faffing about on the sidelines.
He needs to have a bit more confidence in himself, she is clearly desperate to keep him and is behaving in more and more bizarre ways to do so. He needs to know that he CAN tell her I have kids and a DW, and I will see them, like it or lump it.
Agree with WWIFN on using the Inlaws.. excellent idea!